In a dusty backroom off Stoke Newington High Street, north London, Paper Cinema rehearse their latest work, a feature-length rendition of The Odyssey. The group's performance method is strange: unforgettable once seen, but tricky to explain. "I sometimes refer to it as illustrated song or an exploded comic book," says artistic director Nic Rawling. "People have likened it to Japanese shadow theatre or finger puppetry, but it's very different."

Sometimes, he tells people the work is ineffable (and that they should see it for themselves, "hint, hint") but let's give description a go. The outfit has five core members: three musicians, who sit amid a variety of instruments, and beside them, in full view of the audience, two puppeteers who manipulate figures and objects and items of scenery, cut out of flat card and mostly monochrome, in front of a digital video camera. The results are broadcast on a screen behind and this combination – dreamy score, projected image with flat objects moving in and out of focus – works out like a live-made Disney film. Watching the performers, heads down, working to a precise choreography, is as engrossing as the story being projected above them.

"In early gigs," says guitarist and composer Chris Reed, "we'd play behind the audience." But they started to get a better reaction when "the working parts" – as puppeteer Imogen Charleston calls the team – "were out there to see". In recent years, they've played in venues as diverse as the Tate, sharing space with John Martin paintings, and outside Notre Dame, projecting on to a sheet between two trees. They regularly play in the north-east London rehearsal room (an event space in its own right, wryly known as Stoke Newington International Airport) and this month they will play their first West End gig, at the Criterion as part of the theatre's "Playing the Games" season.

The troupe was founded by Rawling eight years ago, initially playing as a sort of human art installation, improvising puppeteered films for events such as Down the Rabbit Hole at the Oxo Tower. Their first narrative piece, commissioned by Battersea Arts Centre, was a 10-minute short featuring 52 puppets. Their next work was 18-minutes with 85 pieces. The Odyssey (another co-production with the BAC) is 70 minutes and they haven't dared count the puppets. "It would scare the pants off me," says Rawling. "We're always seeking new places to stuff them."

Keyboardist Ed Dowie and violinist Quinta make up the five-piece, but Rawling credits a wider family of helpers for the difficult part – cutting out the card needed for a show. "We've got a great bunch of mates," he says, "who get hoiked in before a new performance. We hand them free wine and a scalpel."